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\title{Math 120: Homework 7}
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\begin{itemize}[noitemsep]
  \item Section 7.1 \# 12
  \item Section 7.3 \# 22, 34
  \item Section 7.4 \# 4
\end{itemize}

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{\textbf{Problem 7.1 \#12.}} Prove that any subring of a field containing
$1$ is an integral domain.

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{\textbf{Problem 7.3 \#22.}} Let $a$ be an element of the ring $R$.

(a) Prove that $\{ x \in R|a x = 0 \}$ is a right ideal and $\{ y \in R|y a =
0 \}$ is a left ideal (called respectively the right and left
{\textit{annihilators}} of $a$ in $R$.

(b) Prove that if $L$ is a left ideal of $R$ then $\left\{ x \in R| \text{$x a
= 0$ for all $a \in L$} \right\}$ is a two-sided ideal (called the left
{\textit{annihilator}} of $L$ in $R$).


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{\textbf{Problem 7.3 \#34.}} Let $I$ and $J$ be ideals of $R$.

(a) Prove that $I + J$ is the smallest ideal of $R$ containing both $I$ and
$J$.

(b) Prove that $I J$ is an ideal contained in $I \cap J$.

(c) Give an example where $I J \neq I \cap J$.

(d) Prove that if $R$ is comutative and if $I + J = R$ then $I J = I \cap J$.

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{\textbf{Problem 7.4 \#4.}} Assume that $R$ is commutative. Prove that $R$
is a field if and only if $0$ is a maximal ideal.

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